( 852 ) 
A, to P, the length of the radius vector A,Q and likewise that of 
any radius vector 7Q (if 7 is an arbitrary point already passed) 
increases continually. 
From this we conclude in the first place that between A, and P 
the curve r cannot meet itself and then that, when pursuing 7 from 
A, certainly a point B, ts arrived at, possessing a distance 8 from A,. 
For, if such a point were never reached, we could point out on 
r a series of points 
ER EE TALE ORK Co Sree rae Cee 
which would not end at any number « of the second class of 
numbers, which points would possess from A, the distances 
Er 9 Ege ee + Ens El sy. + ++ Exgeees 
which would continually increase in this order, but remain smaller 
than 8. This however is impossible because the set of the differences 
Eat — Ez 
must be denumerable. 
Any point A, is followed by a well-determined point 4,; such an 
arc A,B, we shall call a g-arc; the distance between the end points 
of a B-are is B; its length lies between 3 and AV 2, as is easily seen. 
Now in the first place it can occur that the pursuing branch and 
the recurrent one after a finite number of g-ares have met either 
each other or one of the two itself. 
In that case we possess a closed single tangent curve to the vector 
distribution. 
If not, the pursuing branch and the recurrent one can be continued 
over an infinite number of g-ares without a meeting taking place; 
this case we shall investigate more closely. 
Let y be = 1/,6 and let A,,A,,A,,.... be a series of points on 
r in such a way, that each are A,A,41 is a y-arc. We are now 
sure that this series of points can reach each finite index. Let farther- 
on p be an integer greater than the quotient of the spherical surface 
by the surface of a eircle of radius */,y on the sphere. 
Let us pursue 7 from A, and let us describe round each 
point A, a circle with radius */,y, then two consecutive ones of those 
circles touch each other on their outside, and any four consecutive 
circles lie entirely outside each other ; however, when A, is reached, two 
circles intersecting each other must have appeared, and at the same 
time or already before, a first point F on r must have been reached 
possessing a distance y from a point G lying more than a g-arc behind 
it. If G lies between Aj and A,, then Aj;4; and Aj4o have a 
distance from G which is >y, whilst A; is separated by less than _ 
